Part of the final exam requirement for each course in the MASS program is the presentation of a topic chosen by the student, researched by him or her in advance. Here are some of the topics students have presented in their final exams.

Linear Algebra in Geometry

Stories about Research in MASS

Director of MASS Program, A. Kouchnirenko, presents David Dumas an award for his research project, MASS-96

As a student in the MASS-96 program at Penn State, I was presented with a number of challenging problems in the three principal areas of coursework that were intended to spawn in-depth research into these fields on the part of MASS students. One such problem that I studied over the course of the semester involved the trinomial analogue of Pascal's triangle. It was then presented as a problem to show that the center column of the triangle had a particularly simple rational generating function. After some research into combinatorics and generating-functionology, augmented by discussion and consultation with the MASS teaching assistants, I was able to prove the identity in a rather unusual way.

David Dumas, MASS-96currently a graduate student at Harvard University

For my final exam presentation in Geometry, I started to investigate the Hausdorff metric. With the help of Dr. Burago and Eric Johnson, I delved into proving Blaschke's Selection Theorem, which states that given a bounded sequence of convex bodies, there exists a subsequence which converges to a convex body. I presented that topic for my final exam. When I returned to Juniata College the following semester, I was asked to present this topic as part of the series of mathematics seminars given by invited professors. I was honored, and I'm thankful to both Juniata and to the MASS program for the experience and confidence I received.

David Shoenthal, MASS-97currently a graduate student at Penn State

The quadratic (logistic) family ƒ(x, λ) = λx(1 − x) of the maps of the unit interval is probably the most extensively studied model in the whole area of dynamical systems, both rigorously and numerically. Existence of a period 3 orbit for an interval map implies existence of orbits of all other periods. This is a particulat case of the celebrated theorem by A. Sharkovsky which was re-discovered by Li and Yorke who called their paper Period three implies chaos, and thus coined the famous term chaos. In the quadratic family a periodic orbit of period 3 appears at the value of λ = 1 + √8. This was known since 1950's but the original proof by P.J. Marburg uses sophisticated methods of the theory of analytic functions. In his popular textbook Nonlinear dynamics and chaos (p. 363) Steve Strogatz tells the story how he suggested to find this value in his advanced class at MIT and how several (rigorous) solutions, some of them using computer algebra, were produced. I repeated Strogatz's experiment in my MASS dynamical systems class. An Nguyen, a participant in MASS Program from University of Texas, Austin not only found the sacramental value, but he did something apparently not known before: he found (also rigorously) the value at which the second period four orbit appears (the first one appears at λ = 1 + √6). The Nguyen's value of λ is 1 + √(4 + ∛108) ≈ 3.9601… and this is quite possibly the last major bifurcation value which can be expressed in radicals.

Anatole Katok, MASS-96 instructorAn Nguyen is presently a second year graduate student in computer science at Stanford.

The learning experience that I gained by working with Dr. Kouchnirenko showed me how to apply abstract ideas in mathematics to very specific examples, especially in a computer programming environment. I was a bit disappointed that I didn't have time to finish the program entirely, but I was able to hack the code enough to work a few nontrivial problems. The program attempts to find roots to polynomials with two variables using a homotopy proceedure (creating a family of polynomials from the given polynomial and a polynomial with known roots), Newton polyhedrons, convexity, and Newton approximation (using only the linear term / first derivative of the given polynomial). While working with Dr. Kouchnirenko I also learned his very unique style of solving problems, something of which I hope to develop in my own style some day.

Chris Staskewicz, MASS-97currently a graduate student at the University of Utah